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+
+\begin{document}
+
+% header
+\begin{center}
+ {\LARGE {\bf Materials Physics II}\\}
+ \vspace{8pt}
+ Prof. B. Stritzker\\
+ SS 2008\\
+ \vspace{8pt}
+ {\Large\bf Tutorial 3}
+\end{center}
+
+\vspace{8pt}
+
+The specific heat (capacity) is the measure of the energy
+required to increase the temperature of a unit quantity of a substance
+by a certain temperature interval.
+Thus, the specific heat at constant volume $V$ is given by
+\[
+c_V = \frac{\partial u}{\partial T}
+\]
+in which $u$ is the energy density of the system.
+
+\section{Specific heat in the classical theory of the harmonic crystal -\\
+         The law of Dulong and Petit}
+
+In the classical theory of the harmonic crystal equilibrium properties
+can no longer be evaluated by simply assuming that each ion sits quitly at
+its Bravais lattice site {\bf R}.
+From now on expectation values have to be claculated by
+integrating over all possible ionic configurations weighted by
+$\exp(-E/k_{\text{B}}T)$, where $E$ is the energy of the configuration.
+Thus, the energy density $u$ is given by
+\[
+u=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)},
+\qquad \beta=\frac{1}{k_{\text{B}}T}
+\]
+in which $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$
+is the volume elemnt in crystal phase space.
+${\bf u}({\bf R})$ and ${\bf P}({\bf R})$  are the 3N canonical coordinates
+(here: deviations from equlibrium sites)
+and 3N canonical momenta
+of the ion whose equlibrium site is ${\bf R}$.
+\begin{enumerate}
+ \item Show that the energy density can be rewritten to read:
+       \[
+   u=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
+       \]
+ \item Show that the potential contribution to the energy
+       in the harmonic approximation is given by
+       \begin{eqnarray}
+       U&=&U_{\text{eq}}+U_{\text{harm}} \nonumber \\
+       U_{\text{eq}}&=&\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'})
+       \nonumber \\
+       U_{\text{harm}}&=&\frac{1}{2}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
+       [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'})
+       [u_v({\bf R})-u_v({\bf R'})] \nonumber
+       \end{eqnarray}
+       in which
+$\Phi_{\mu v}({\bf r})=
+ \frac{\partial^2 \Phi({\bf r})}{\partial r_{\mu}\partial r_v}$
+       and $\Phi({\bf r})$ is the potential contribution of two atoms
+       separated by ${\bf r}$.
+       {\bf Hint:}
+
+\end{enumerate}
+
+
+\section{Specific heat in the quantum theory of the harmonic crystal -\\
+         Models of Debye and Einstein}
+
+\begin{enumerate}
+ \item
+ \item
+\end{enumerate}
+
+\end{document}